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The question asks which center of dilation could place point \( A' \) at \( (-4, 1) \). We need to determine the correct dilation center based on the given coordinates and transformation.

The key observation is that point \( A \) at \( (-2, 3) \) is transformed to \( A'(-4, 1) \). To find the correct center of dilation, we will:

1. Calculate the slope of the line connecting the center of dilation (denoted as \( C(x, y) \)) to both \( A \) and \( A' \).
2. Compare the distance ratio between \( A \) and \( A' \), which will help identify the center that would maintain the correct proportions.

### Step-by-step approach:
- The line connecting point \( A(-2, 3) \) and \( A'(-4, 1) \) forms a straight line.
- To find the center of dilation, it must lie on this same line.
- By calculating the slopes from each center option to both points and comparing with the transformation ratios, we identify the correct center.

After analyzing the points, the center of dilation that correctly maps \( A \) to \( A' \) is at \( (-3, -2) \).

Would you like further details on how the slope and dilation factor were determined?

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### 5 Related Questions:
1. What is the formula to calculate the dilation factor between two points?
2. How can we find the center of dilation if given two corresponding points?
3. What happens to the other points of the figure when dilated from this center?
4. How does the distance ratio between corresponding points change with different centers?
5. How can we confirm if a dilation is an enlargement or a reduction?

### Tip:
When dealing with dilations, always check both the distance ratios and slopes between points to ensure you find the correct center and scale factor.

What center of dilation could place A' at (-4, 1)?

Learn how to find the center of dilation for transforming point A at (-2, 3) to A' at (-4, 1). This problem explores key geometry concepts including dilation, scaling, and coordinate geometry. Suitable for high school students in Grades 9-11.

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